If $f(x), g(x)$ and $h(x)$ are three polynomials of degree 2, then $\begin{vmatrix}f(x) &g(x) &h(x)\\f'(x) &g'(x) &h' (x)\\f"(x)& g''(x) &h''(x)\end{vmatrix}$ is a polynomial of degree
Answer & explanation
Correct answer: option 4
Since $f(x), g(x)$ and $h(x)$ are polynomials of degree 3.
$∴f"'(x)=g'''(x)=h'''(x) = 0$.
Now,
$\phi'(x)=\begin{vmatrix}f'(x) &g'(x) &h'(x)\\f'(x) &g'(x) &h' (x)\\f''(x)& g''(x)& h''(x)\end{vmatrix}+\begin{vmatrix}f(x) &g(x) &h(x)\\f''(x) &g''(x) &h'' (x)\\f''(x)& g''(x)& h''(x)\end{vmatrix}+\begin{vmatrix}f(x) &g(x) &h(x)\\f'(x) &g'(x) &h' (x)\\f'''(x)& g'''(x)& h'''(x)\end{vmatrix}$
$⇒\phi'(x)=0+0+\begin{vmatrix}f(x) &g(x) &h(x)\\f'(x) &g'(x) &h' (x)\\0&0&0\end{vmatrix}=0$
$∴\phi'(x)$ = Constant
⇒ $\phi'(x)$ → Polynomial of degree one